Technically, the feeling functions are not feelings, emotions, or actual experiences themselves. These functions are used to understand emotions, internally and externally, and to make judgements as to how they fit into a higher framework. For Fi, the framework is that of moral standards. And for Fe, the framework is that of ethical standards.
Emotions themselves, I would agree, are intertwinded into every aspect of life and give important depth to our experiences. Our personal experiences affect the way we use our functions and shapes what would be our identity. The feeling functions alone do not determine our identity; however, the way we use those functions gives indications as to what lies within our core selves.
Nature & Nurture - with the MBTI cognitive function model as our nature, and our unique life background as our nurture. Both separate, but intertwined and equally as important.